Applied Statistics and Probability for Engineers

14-7 2kFACTORIAL DESIGNS 525

Similarly, the main effect of Bis found by averaging the observations on the top of the square,

where Bis at the high level, and subtracting the average of the observations on the bottom of

the square, where Bis at the low level:

 (14-11)

1

2 n

3 aabb 1124



aab

2 n



b 112

2 n

AyAyA

 (14-12)

1

2 n

3 baba 1124



bab

2 n



a 112

2 n

ByByB

 (14-13)

1

2 n

3 ab 112 ab 4

AB

ab 112

2 n



ab

2 n

Table 14-12 Signs for Effects in the 2^2 Design

Treatment Factorial Effect

Combination IABAB

112 

a 

b 

ab 

Finally, the ABinteraction is estimated by taking the difference in the diagonal averages

in Fig. 14-12, or

The quantities in brackets in Equations 14-11, 14-12, and 14-13 are called contrasts.For

example, the Acontrast is

ContrastAaabb 112

In these equations, the contrast coefficients are always either 1 or1. A table of plus and

minus signs, such as Table 14-12, can be used to determine the sign on each treatment

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